OFFSET
0,3
COMMENTS
A multiset is uniform if all multiplicities are equal. A multiset partition is uniform if all parts have the same size.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..100
FORMULA
a(n) = 4 <=> n in { A000040 }. - Alois P. Heinz, Feb 03 2022
EXAMPLE
The a(1) = 1 though a(6) = 48 multiset partitions:
{1} {11} {111} {1111} {11111} {111111}
{12} {123} {1122} {12345} {111222}
{1}{1} {1}{1}{1} {1234} {1}{1}{1}{1}{1} {112233}
{1}{2} {1}{2}{3} {11}{11} {1}{2}{3}{4}{5} {123456}
{11}{22} {111}{111}
{12}{12} {111}{222}
{12}{34} {112}{122}
{13}{24} {112}{233}
{14}{23} {113}{223}
{1}{1}{1}{1} {122}{133}
{1}{1}{2}{2} {123}{123}
{1}{2}{3}{4} {123}{456}
{124}{356}
{125}{346}
{126}{345}
{134}{256}
{135}{246}
{136}{245}
{145}{236}
{146}{235}
{156}{234}
{11}{11}{11}
{11}{12}{22}
{11}{22}{33}
{11}{23}{23}
{12}{12}{12}
{12}{12}{33}
{12}{13}{23}
{12}{34}{56}
{12}{35}{46}
{12}{36}{45}
{13}{13}{22}
{13}{24}{56}
{13}{25}{46}
{13}{26}{45}
{14}{23}{56}
{14}{25}{36}
{14}{26}{35}
{15}{23}{46}
{15}{24}{36}
{15}{26}{34}
{16}{23}{45}
{16}{24}{35}
{16}{25}{34}
{1}{1}{1}{1}{1}{1}
{1}{1}{1}{2}{2}{2}
{1}{1}{2}{2}{3}{3}
{1}{2}{3}{4}{5}{6}
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[Length[Select[mps[m], SameQ@@Length/@#&]], {m, Table[Join@@Table[Range[n/d], {d}], {d, Divisors[n]}]}], {n, 8}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 26 2018
EXTENSIONS
More terms from Alois P. Heinz, Jan 30 2019
Terms a(14) and beyond from Andrew Howroyd, Feb 03 2022
STATUS
approved